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We consider the problem of counting the number of varieties in a family over Q with a rational point. We obtain lower bounds for this counting problem for some families over P1, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher-dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce definitions for the integrated equivariant Milnor number μIG and the equivariant Milnor class ℳG(Z), for singular hypersurfaces. We prove that the μIG are constant on the strata in a Whitney stratification of Z, along with the correlation ℳG(Z) = ℳG,0(Z) = 1 |G|∑i=1kμ IG(x i) for hypersurfaces hosting isolated singularities x1,…,xk, where ℳG,0(Z) denotes the 0th equivariant Milnor class of Z.We also introduce the equivariant Fulton–Johnson class of singular hypersurfaces. We give an equivariant version of Verdier’s specialization morphism in homology, and also for constructible functions. This is used for finding a relation between equivariant Fulton–Johnson and Schwartz–MacPherson classes. [ABSTRACT FROM AUTHOR]

For any complete C -algebraic variety X and its associated compact C -analytic space X , it follows from the well known GAGA principle that the algebraic Picard group P i c (X) and the analytic Picard group P i c (X) are isomorphic. Our main purpose here is to investigate an analogous situation for non complete situation, namely C -algebraic affine hypersurfaces. [ABSTRACT FROM AUTHOR]

For a germ of a variety V,0⊂CN,0$\mathcal {V}, 0 \subset \mathbb {C}^N, 0$, a singularity V0$\mathcal {V}_0$ of "type V$\mathcal {V}$" is given by a germ f0:Cn,0→CN,0$f_0: \mathbb {C}^n, 0 \rightarrow \mathbb {C}^N, 0$, which is transverse to V∖{0}$\mathcal {V}\setminus \lbrace 0\rbrace$ in an appropriate sense, such that V0=f0−1(V)$\mathcal {V}_0 = f_0^{-1}(\mathcal {V})$. In part I of this paper, we introduced for such singularities the Characteristic Cohomology for the Milnor fiber (for V$\mathcal {V}$ a hypersurface), and complement and link (for the general case). It captures the cohomology of V0$\mathcal {V}_0$ inherited from V$\mathcal {V}$ and is given by subalgebras of the cohomology for V0$\mathcal {V}_0$ for the Milnor fiber and complements, and is a subgroup for the cohomology of the link. We showed these cohomologies are functorial and invariant under diffeomorphism groups of equivalences KH$\mathcal {K}_{H}$ for Milnor fibers and KV$\mathcal {K}_{\mathcal {V}}$ for complements and links. We also gave geometric criteria for detecting the nonvanishing of the characteristic cohomology. In this paper, we apply these methods in the case V$\mathcal {V}$ denotes any of the varieties of singular m×m$m \times m$ complex matrices, which may be either general, symmetric, or skew‐symmetric (with m$m$ even). For these varieties, we have shown in another paper that their Milnor fibers and complements have compact "model submanifolds" for their homotopy types, which are classical symmetric spaces in the sense of Cartan. As a result, we first give the structure of the characteristic cohomology subalgebras for the Milnor fibers and complements as images of exterior algebras (or in one case a module on two generators over an exterior algebra). For links, the characteristic cohomology group is the image of a shifted upper truncated exterior algebra. In addition, we extend these results for the complement and link to the case of general m×p$m \times p$ complex matrices. Second, we then apply the geometric detection methods introduced in Part I to detect when specific characteristic cohomology classes for the Milnor fiber or complement are nonzero. We identify an exterior subalgebra on a specific set of generators and for the link that it contains an appropriate shifted upper truncated exterior subalgebra. The detection criterion involves a special type of "kite map germ of size ℓ$\ell$" based on a given flag of subspaces. The general criterion that detects such nonvanishing characteristic cohomology is then given in terms of the defining germ f0$f_0$ containing such a kite map germ of size ℓ$\ell$. Furthermore, we use a restricted form of kite spaces to give a cohomological relation between the cohomology of local links and the global link for the varieties of singular matrices. [ABSTRACT FROM AUTHOR]

We establish the non-commutative analogue of Grothendieck's standard conjecture D for the differential graded category of G-equivariant matrix factorizations associated to an isolated hypersurface singularity where G is a finite group. [ABSTRACT FROM AUTHOR]

We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood‐type inequality and we characterize the corresponding maximal representations. [ABSTRACT FROM AUTHOR]

Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito's theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (f, X), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the concept of ρ -regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor tube fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map G with respect to the levels of a function ρ , which guarantees ρ -regularity. Consequently, both conditions are crucial for the presence of fibration structures. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts. [ABSTRACT FROM AUTHOR]

The number of Morse points in a Morsification determines the topology of the Milnor fibre of a holomorphic function germ f with isolated singularity. If f has an arbitrary singular locus, then this nice relation to the Milnor fibre disappears. We show that despite this loss, the numbers of stratified Morse singularities of a general linear Morsification are effectively computable in terms of topological invariants of f. [ABSTRACT FROM AUTHOR]

Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber F can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit formulas are still lacking in full generality, even for b1(F). We study here the "generic" case (in which b1(F) is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations with trivial algebraic monodromy. Our main focus is on the cohomology jump loci and the lower central series quotients of π1(F). In the process, we produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet non-isomorphic fundamental groups: the difference is picked by the higher-depth characteristic varieties and by the Schur multipliers of the second nilpotent quotients. [ABSTRACT FROM AUTHOR]

In Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps f : (R n , 0) → (R k , 0) , where n ≥ k ≥ 2 , with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class C ℓ , ℓ ≥ 2 , with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class C ℓ between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations. [ABSTRACT FROM AUTHOR]

For any closed orientable 3‐manifold, there is a volume function defined on the space of all Seifert representations of the fundamental group. The maximum absolute value of this function agrees with the Seifert volume of the manifold due to Brooks and Goldman. For any Seifert representation of a graph manifold, the authors establish an effective formula for computing its volume, and obtain restrictions to the representation as analogous to the Milnor–Wood inequality (about transversely projective foliations on Seifert fiber spaces). It is shown that the Seifert volume of any graph manifold is a rational multiple of π2$\pi ^2$. Among all finite covers of a given nongeometric graph manifold, the supremum ratio of the Seifert volume over the covering degree can be a positive number, and can be infinite. Examples of both possibilities are discovered, and confirmed with the explicit values determined for the finite ones. [ABSTRACT FROM AUTHOR]

This is a survey article, in which we explore how the presence of singularities affects the geometry and topology of complex projective hypersurfaces. [ABSTRACT FROM AUTHOR]

Let f : (ℝ n , 0) → (ℝ , 0) be a definable function germ of class 2 and let (X , 0) ⊂ (ℝ n , 0) be a germ of a closed definable set. We investigate topological invariants associated with f | X . In particular, we give several topological formulae for the Euler characteristics of related sets. We also relate the topology of f | X to the topology of a definable function with isolated critical point in the stratified case. [ABSTRACT FROM AUTHOR]

By applying an inverse Kaluza-Klein procedure, we provide explicit coordinate expressions for Riemannian metrics on two homeomorphic but not diffeomorphic spheres in seven dimensions. We identify Milnor's bundles, among which ten out of the fourteen exotic seven-spheres appear (ignoring orientation), with non-principal bundles having homogeneous fibres. Then, we use the techniques in [1] to obtain a general ansatz for the coordinate expression of a metric on the total space of any Milnor's bundle. The ansatz is given in terms of a metric on S4, a metric on S3 (which can smoothly vary throughout S4), and a connection on the principal SO(4)-bundle over S4. As a concrete example, we present explicit formulae for such metrics for the ordinary sphere and the Gromoll-Meyer exotic sphere. Then, we perform a non-abelian Kaluza-Klein reduction to gravity in seven dimensions, according to (a slightly simplified version of) the metric ansatz above. We obtain the standard four-dimensional Einstein-Yang-Mills system, for which we find solutions associated with the geometries of the ordinary sphere and of the exotic one. The two differ by the winding numbers of the instantons involved. [ABSTRACT FROM AUTHOR]

Let BG be the classifying space of an algebraic group G over the field ${\mathbb C}$ of complex numbers. There are smooth projective approximations X of $BG\times {\mathbb P}^{\infty}$ , by Ekedahl. We compute a new stable birational invariant of X defined by the difference of two coniveau filtrations of X , by Benoist and Ottem. Hence we give many examples such that two coniveau filtrations are different. [ABSTRACT FROM AUTHOR]

This document is a foreword for a volume celebrating two prominent Romanian mathematicians, Laurentiu Paunescu and Alexander Suciu. The contributions to the volume were delivered by speakers and participants in a conference held in their honor in Bucharest. The foreword provides brief biographical information about both mathematicians, including their educational background, research interests, and academic achievements. It also mentions their personal lives and interests outside of mathematics. [Extracted from the article]

In previous work jointly with Geske, Maxim, and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of (one variable) Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this paper, we give a positive answer to that question, which implies that the direct sum decomposition of the torsion part of Alexander modules into generalized eigenspaces is in fact a decomposition of MHS. We also show that the MHS on the generalized eigenspace of eigenvalue |$1$| can be constructed without passing to a suitable finite cover (as is the case for the MHS on the torsion part of the Alexander modules), and compute it under some purity assumptions on the base space. Further, we show a formula relating the Alexander module's Hodge numbers to those of finite covers of the base space, under some assumptions. Dedicated to the memory of Georgia Benkart [ABSTRACT FROM AUTHOR]

Let (X, 0) be the germ of either a subanalytic set X⊂Rn$X \subset {\mathbb {R}}^n$ or a complex analytic space X⊂Cn$X \subset {\mathbb {C}}^n$, and let f:(X,0)→(Kk,0)$f: (X,0) \rightarrow ({\mathbb {K}}^k, 0)$ be a K${\mathbb {K}}$‐analytic map‐germ, with K=R${\mathbb {K}}={\mathbb {R}}$ or C${\mathbb {C}}$, respectively. When k=1$k=1$, there is a well‐known topological locally trivial fibration associated with f, called the Milnor–Lê fibration of f, which is one of the main pillars in the study of singularities of maps and spaces. However, when k>1$k>1$ that is not always the case. In this paper, we give conditions which guarantee that the image of f is well‐defined as a set‐germ, and that f admits a Milnor–Lê fibration. We also give conditions for f to have the Thom property. Finally, we apply our results to mixed function‐germs of type fg¯:(X,0)→(C,0)$f \bar{g}: (X,0) \rightarrow ({\mathbb {C}},0)$ on a complex analytic surface X⊂Cn$X \subset {\mathbb {C}}^n$ with arbitrary singularity. [ABSTRACT FROM AUTHOR]

We give an example of a nonzero odd degree element of the classifying space of a connected Lie group such that all higher Milnor operations vanish on it. It is a counterexample of a conjecture of Kono and Yagita [Trans. Amer. Math. Soc. 339 (1993), pp. 781–798]. [ABSTRACT FROM AUTHOR]

For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$. We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge–Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements. [ABSTRACT FROM AUTHOR]

Local system cohomology groups of the complements of hyperplane arrangements have played an important role in the theory of hypergeometric integrals, the topology of Milnor fibers and covering spaces. One of the important theorems is the vanishing theorem for generic ℂ -local systems which goes back to Aomoto's work. Later, Cohen, Dimca, and Orlik proved a stronger version of the vanishing theorem. In this paper, we prove a Cohen–Dimca–Orlik type theorem for ℤ -local systems. [ABSTRACT FROM AUTHOR]

Copyright of Annales Mathematiques Blaise Pascal is the property of Laboratoire mathematiques Blaise Pascal - CNRS UMR 6620 and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Over fields of characteristic zero, we construct equivalences between certain categories of bialgebras which are generated by grouplikes and generalized primitives, and certain categories of structured Lie algebras. The relevant families of bialgebras include many which are not connected, and which fail to admit antipodes. [ABSTRACT FROM AUTHOR]

It is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots , f^{k_0}$ if $\{f^{k_1}=\cdots =f^{k_m}=0\}$ is a nondegenerate complete intersection variety for any $k_1,\ldots ,k_m\in \{1,\ldots , k_0\}$. [ABSTRACT FROM AUTHOR]

In this paper we present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases. This allows us to compare our results with the previous ones. [ABSTRACT FROM AUTHOR]

We define the Milnor number of a one-dimensional holomorphic foliation F as the intersection number of two holomorphic sections with respect to a compact connected component C of its singular set. Under certain conditions, we prove that the Milnor number of F on a three-dimensional manifold with respect to C is invariant by -1 topological equivalences. [ABSTRACT FROM AUTHOR]

Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial 2-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent. [ABSTRACT FROM AUTHOR]

We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part the question concerning the motivic nearby cycles of restriction functions in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci. [ABSTRACT FROM AUTHOR]

Parametrized motion planning algorithms [1] have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles. Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values. [ABSTRACT FROM AUTHOR]

A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of a noncommutatively stably semiorthogonally indecomposable (NSSI) scheme. This property implies, among other things, that each connected closed subscheme has indecomposable derived category of coherent sheaves and that if Y is NSSI, then for any variety X all semiorthogonal decompositions of X x Y are induced from decompositions of X. We prove that any scheme which admits an affine morphism to an abelian variety is NSSI and that the total space of a fibration over a NSSI base with NSSI fibers is also NSSI. We apply this indecomposability to deduce that there are no phantom subcategories in some varieties, including surfaces C x ℙ¹, where C is any smooth proper curve of positive genus. [ABSTRACT FROM AUTHOR]

We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of A∞$A_\infty$‐structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an n$n$‐dimensional affine hypersurface of degree n+2$n+2$ and the double cover of the n$n$‐dimensional affine space branched along a degree 2n+2$2n+2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel (Proceedings of the International Congress of Mathematicians, 2002) which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory. [ABSTRACT FROM AUTHOR]

We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it to compute explicitly the lowest (possibly nontrivial) vanishing cohomology group of the Milnor fiber. [ABSTRACT FROM AUTHOR]

We prove that for two germs of analytic mappings f , g : (C n , 0) → (C p , 0) with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family { f t } of analytic maps with f 0 = f , f 1 = g which has a so-called uniform stable radius for the Milnor fibration. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is a generalization of the result by C. Bivia-Ausina. Consequently, we obtain that the Milnor number of a non-degenerate isolated complete intersection singularity is an invariant of Newton boundaries. [ABSTRACT FROM AUTHOR]

We describe the topology of the local Milnor fiber of a function f defined on a 3-dimensional subanalytic subset W ⊂ R 3 , in terms of the embedded topological type of its link K f . Precisely, we prove that the interior of the local Milnor fiber of ‖ f ‖ is homeomorphic to the complement of K f in the corresponding sphere, extending a result due to Milnor to non-analytic situations. We also prove that the topology of the fiber does not change if we use a suitable fundamental system of neighborhoods instead of balls, extending to the real setting a classical result due to Lê and Teissier for complex functions. [ABSTRACT FROM AUTHOR]

For a complex central essential hyperplane arrangement |$\mathcal{A}$|⁠ , let |$F_{\mathcal{A}}$| denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of |$F_{\mathcal{A}}$|⁠. We prove that the map taking an arrangement |$\mathcal{A}$| to the Hodge-Deligne polynomial of |$F_{\mathcal{A}}$| is locally constant on the realization space of any loop-free matroid. When |$\mathcal{A}$| consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of any simple normal crossing compactification of |$F_{\mathcal{A}}$|⁠. As a direct consequence, we obtain a combinatorial formula for the top weight cohomology of |$F_{\mathcal{A}}$|⁠ , recovering a result of Dimca and Lehrer. [ABSTRACT FROM AUTHOR]

Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the reduced cohomology of the Milnor fiber in lowest possible non-trivial degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant. [ABSTRACT FROM AUTHOR]

In this article we study a conjecture of Geiss-Leclerc-Schröer, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type A 2 quiver. In the general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic. [ABSTRACT FROM AUTHOR]

Let $f\,{:}\,(\mathbb R^n,0)\to (\mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F\,{:}\, (\mathbb{R}^{1+n},0) \to (\mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) \cap B_\epsilon^n$ , $0 , be the "generalized" Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui. [ABSTRACT FROM AUTHOR]

A log generic hypersurface in P n with respect to a birational modification of P n is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic hypersurface is defined similarly but restricting to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product f = f 1 ... f p of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple (f 1 , ... , f p , g) and for the product fg, if g is log generic. We also show that the stronger version of the monodromy conjecture, relating the motivic zeta function with the Bernstein–Sato ideal, holds for the tuple (f 1 , ... , f p , g) and for the product fg, if g is log very-generic. Even the case f = 1 is intricate, the proof depending on nontrivial properties of Bernstein–Sato ideals, and it singles out the class of log (very-) generic hypersurfaces as an interesting class of singularities on its own. [ABSTRACT FROM AUTHOR]

We study the equisingularity of a family of function germs { f t : (X t , 0) → (C , 0) } , where (X t , 0) are d-dimensional isolated determinantal singularities. We define the (d - 1) th polar multiplicity of the fibers X t ∩ f t - 1 (0) and we show how the constancy of the polar multiplicities is related to the constancy of the Milnor number of f t and the Whitney equisingularity of the family. [ABSTRACT FROM AUTHOR]

In this paper we study relationships between some topological and analytic invariants of zero-dimensional germs, or multiple points. Among other things, it is shown that there exist no rigid zero-dimensional Gorenstein singularities and rigid almost complete intersections. In the proof of the first result we exploit the canonical duality between homology and cohomology of the cotangent complex, while in the proof of the second we use a new method which is based on the properties of the torsion functor. In addition, we obtain highly efficient estimates for the dimension of the spaces of the first lower and upper cotangent functors of arbitrary zero-dimensional singularities, including the space of derivations. We also consider examples of nonsmoothable zero-dimensional noncomplete intersections and discuss some properties and methods for constructing such singularities using the theory of modular deformations, as well as a number of other applications. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of the moduli of elliptically fibred K3 surfaces and is closely related to the root lattices of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positivedimensional ambi-typical strata, that is, strata which are both Shimada root strata and monodromy strata. We also discuss the relationship with moduli spaces of latticepolarised K3 surfaces. The appendix by M. Kirschmer contains computational results about the 1-dimensional ambi-typical strata. [ABSTRACT FROM AUTHOR]

We introduce several sufficient conditions to guarantee the existence of the Milnor vector field for new classes of singularities of map germs. This special vector field is related with the equivalence problem of the Milnor fibrations for real and complex singularities, if they exit. [ABSTRACT FROM AUTHOR]

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A'Campo's theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety. [ABSTRACT FROM AUTHOR]